PARIS — New geometry — linear and circular cutting — rules the runways for the fall/winter Paris season. From the Issey Miyake math lesson through the Rick Owens asymmetrics to Vionnet’s folded squares, sharp cutting and soft folds are the power pieces.

The interlacing black nets at the Issey Miyake show on Friday might have been a set for a sports event — but the designer Dai Fujiwara had a more complex answer. The structure, like the collection’s colorful knits, were designed according to the geometric theories of William Thurston, the U.S. mathematician who took a bow with the designer at the end of the show.

Models coming to the catwalk from different directions wore loops of knitting in green, mauve, pink and apricot, wound around simple tops above shiny black draped pants. Those outfits were followed by jackets that had been cut on the curve with spiraling piping to underscore the fluidity.

The compass may have played a bigger role than the ruler in this show, for everything seemed to work in the round, even a black leather coat with whorls of stitching and hose covered in a pattern of watery globules.

Citing everything from the Milky Way to a doughnut, Mr. Fujiwara made straight angle tailoring credible, as well as the rounded version of the wind coats that Mr. Miyake invented three decades ago. And you did not need a top grade in math to understand the fundamentals of this thought-provoking Issey Miyake show: clean geometric lines with imaginative embellishment, like the squares of organza appliqued with stars for a celestial moment.

The greatest transformations came at Vionnet, where the designer Rodolfo Paglialunga seems to have got the hang of this historic house. That means every draped cocktail dress in liquid satin and soft lace seems to be built from flat fabric on to a curving, womanly body. Add sweet colors revealed delicately behind black draping and you have the house signature — literally, when Madame Vionnet’s tiny emblem was part of a red and white square pattern.

But that was nothing compared to the setting in the former home of Jean Cocteau, where Picasso’s phone number was scribbled on a slate and Pierre le Tan’s drawings were on the walls.

It has been a long haul for Matteo Marzotto to resuscitate the Vionnet brand, but he is wise enough to take it step by step. A new departure was a trim cashmere coat and a caped knit. And everything in this small presentation suggested that the brand is being well nurtured.

The Japanese designer known as Sacai is making a name for her sweetness and subtlety. When is a Chanel suit not the familiar tweed two-piece, but flowing with imagination? Sacai created a knitted sleeveless jacket and used the same woolly knit for the sleeves of a wispy blouse. Then, the same chiffon material from the blouse peeked out of the open seams in the outfit’s skirt.

The designer does the currently fashionable mix of fabrics for an intensely womanly look, as in a using a nylon parka material for a full skirt. But instead of the deconstructed look that an older Japanese generation began half a century ago, this show was about reconstruction in the most delightful way.

Is it possible for a collection to be futuristic, cool and wearable — all at the same time?

Rick Owenspulled off that top trio for the autumn 2010 season by keeping the wilder elements — dark cat’s eye make- up, geometrically patterned hose and the designer’s always strange footwear — to the extremities. The collection focused on jackets, and only when they were teamed with asymmetric skirts falling to a point did they seem even slightly weird.

“It’s always a sect — I wanted them to look strong and disciplined, like glamorous nuns,” said Mr. Owens, who has perfected his look of a powerful woman, but not that “power woman” associated with wearing masculine pantsuits to work.

The core of the collection was in sleeveless coats that also served as dresses, while the bared arms were finished with gloves embellished with spirals of fur, not least at the fingertips. Other power decoration came as appliqued squares of horn. Because shades of gray dominated over black, the collection had a lightness to it. The triangular skirts gave a freedom to striding legs. And Mr. Owens achieved that rare effect: making his collection look avant garde and chic at the same time.

When Carey Mulligan, the young Oscar-nominated star of the indie movie “An Education,” wore a Nina Ricci dress to the Golden Globes, it was a coup for Peter Copping, the new designer at the historic Parisian house. That dress was a strapless navy blue column with intricate workmanship and would have been a good line to follow.

Perhaps the movie, set in the early 1960s, just as sexual freedom was burgeoning in England, inspired the big hair that seems to be a new thing on the current runways. But that look killed the delicacy of Mr. Copping’s vision, which is based on the pretty lingerie looks that he did well last season. Translate lace into flowers and what do you get — a bunch of roses at the waist or making a ball out of a slim skirt.

There were a lot of good things about the collection: fine tailoring, not least in a long coat over a pretty evening dress. The designer has a real feel for the lingerie softness of satin and a dusty pale palette. But it is indeed an education to learn how to bring a historic brand forward. Mr. Copping has all the elements. Now he needs to meld them.